On the stability of homogeneous Einstein manifolds II

Abstract

For any G $G$ -invariant metric on a compact homogeneous space M = G / K $M=G/K$ , we give a formula for the Lichnerowicz Laplacian restricted to the space of all G $G$ -invariant symmetric 2-tensors in terms of the structural constants of G / K $G/K$ . As an application, we compute the G $G$ -invariant spectrum of the Lichnerowicz Laplacian for all the Einstein metrics on most generalized Wallach spaces and any flag manifold with b 2 ( M ) = 1 $b_2(M)=1$ . This allows to deduce the G $G$ -stability and critical point types of each of such Einstein metrics as a critical point of the scalar curvature functional.